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\title{REPORT}
\author{Daniel Zawada}
\date{May 30th 2011}

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The homegenization problem for the MFPT inside a sphere is given by a diffusion equation with robin boundary conditions,

\begin{equation}
	\Delta v = \frac{-1}{D}, \hspace{5mm} v \in \Omega ; \hspace{5mm} \epsilon \partial_r v + \kappa v = 0,\hspace{5mm} v \in \partial \Omega
\end{equation}

Where $v=v(r)$ is the MFPT, $\epsilon$ is the radius of each trap, and $\kappa$ is a geometric factor determined by the type of traps and their placement. This equation has solution,

\begin{equation}
	\label{eqn:vhomeg}
	v = \frac{-r^2}{6 D} + \frac{1}{6 D} + \frac{1}{3 D} \frac{\epsilon}{\kappa}
\end{equation}

Define the average MFPT in the sphere as $\bar{v} = \frac{1}{|\Omega|} \int v dx$ where $|\Omega|$ is the volume of the sphere and the integral
is over the entire volume. Thus the average MFPT for $v$ given by (\ref{eqn:vhomeg}) is,

\begin{equation}
	\label{eqn:vbarhomeg}
	\bar{v} = \frac{1}{15 D} + \frac{1}{3 D} \frac{\epsilon}{\kappa} 
\end{equation}

For $N$ traps, $\bar{v}$ was calculated asymptotically to be,

\begin{equation}
	\label{eqn:vbarasympt}
	\bar{v} = \frac{|\Omega|}{4 \epsilon D N} \bigg[1 + \frac{\epsilon}{\pi} \log\bigg(\frac{2}{\epsilon}\bigg) + \frac{\epsilon}{\pi}\bigg(-\frac{9 N}{5} + 2(N-2)\log 2 + \frac{3}{2} + \frac{4}{N} \mathcal{H}(x_1,\dots,x_N)\bigg)\bigg] 
\end{equation}

Where $x_i$'s are the trap locations and $\mathcal{H}(x_1,\dots,x_N)$ is the interaction energy defined by,

\begin{equation}
	\label{eqn:hdef}
	\mathcal{H}(x_1,\dots,x_N) = \sum_{i=1}^N \sum_{j=i+1}^N \bigg(\frac{1}{|x_i-x_j|} - \frac{1}{2} \log |x_i-x_j| - \frac{1}{2} \log(2 + |x_i - x_j|)\bigg)
\end{equation}

We are interested in trap configuration that results in a minimum for $\bar{v}$, this is equivalent to
finding the trap configuration that results in minimum $\mathcal{H}$. We introduce $f$ denoting the fraction of the surface area covered in traps,

\begin{equation}
	\label{eqn:fdef}
	f = \frac{\pi \epsilon^2 N}{4 \pi}
\end{equation}

Holding $f$ constant and in the limit $N \gg 1$, (\ref{eqn:vbarasympt}) has the leading order terms,

\begin{equation}
	\bar{v} \sim \frac{1}{D}\bigg(-\frac{9}{20} - \log 2 + \lim_{N \to \infty} \frac{\mathcal{H}}{N^2}\bigg)
\end{equation}

From (\ref{eqn:vbarhomeg}) we know $\bar{v} \to \frac{1}{15 D}$ as $N \to \infty$, thus $\mathcal{H}$ has the leading order
term,

\begin{equation}
	\mathcal{H} \sim \frac{1}{2} \bigg(\frac{14}{15} - \log 2\bigg) N^2 
\end{equation}

We now attempt to evaluate the sum in (\ref{eqn:hdef}) in the limit $N \gg 1$ by approximating the sum as an integral.  Consider 
one trap located at $(r,\theta, \phi) = (1,0,0)$.  We assume the traps are distributed normally along the sphere except in a small
neighbourhood centered at the trap.  We write the trap density as,

\begin{equation}
	P(\theta,\phi) = \left\{\begin{array}{c c}
		0, & 0 < \theta < \theta_0 \\
		\frac{N}{4 \pi}, & \theta_0 < \theta < \pi 
	\end{array}\right.
\end{equation}

$\theta_0$ can be determined from the condition,

\begin{equation}
	\int_0^{2 \pi} \int_{\theta_0}^\pi P(\theta,\phi) \sin \theta d\theta d\phi = N-1
\end{equation}

Evaluating this integral yields $\cos \theta_0 = 1 - 2/N$.  Now we can approximate the sum as,

\begin{equation}
	\mathcal{H} \frac{N}{2}\approx \int_0^{2 \pi} \int_{\theta_0}^\pi P(\theta,\phi) \bigg(\frac{1}{r(\theta)} - \frac{1}{2} \log r(\theta) - \frac{1}{2} \log (2 + r(\theta))\bigg) \sin \theta d\theta d\phi
\end{equation}

Where we have multiplied by $1/2$ since the interactions are counted twice.  This integral yields,

\begin{equation}
	\label{eqn:Hintegral}
	\begin{split}
		\mathcal{H} \approx & N^2 \frac{1}{2} (1-\log 2) - \frac{1}{4} N^2 \log(1+\sqrt{N}) + \frac{1}{8} N^2 \log N - \frac{1}{4} N^{3/2} \\
	& - \frac{1}{4} N \log N +\frac{1}{4} N \log (\sqrt{N} + 1) - \frac{N}{2} \bigg(\frac{1}{2} - \log 2\bigg)
	\end{split}
\end{equation}

This equation provides reasonable agreement with numerical results for $\mathcal{H}$ but does not seem to converge to them
at large $N$.  We propose that $\mathcal{H}$ has the following form based on (\ref{eqn:Hintegral})

\begin{equation}
\begin{split}
	\mathcal{H} &= b_0 N^2 + b_1 N^2 \log N + b_2 N^2 \log (\sqrt{N} + 1) + b_3 N^{3/2}  \\
	& + b_4 N \log N + b_5 N \log(\sqrt{N} + 1) +  b_6 N \\
\end{split}
\end{equation}



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